James Olaleye 
  
  
b. 
bz b, in the R-space (5) 
b. 
b 
Naturally, in the ARDOVS theory, an element of the left image sees only the natural direction axes, 
therefore, we may represent the conjugate left image element as : 
LP 
T, —|J P a (6) 7 
k P 
Theoretically, vector g (eqn.3) should be equal to vector T, in the R -space, i.e. the vector triangle in Figure 5 shouli 
close as shown in Figure 3c. This implies that the vectors T, and T» should intersect at a point. However, in reality, 
they do not intersect at a point due to some residual parallax as shown in figure 5. Therefore, we employ mathematicil 
optimization strategy to stretch the vectors T; and T» along their direction to a point at which the parallax vector P is oj 
minimum possible length. This is done in the R-space. 
Let P be the parallax vector signifying the want of intersection of the two conjugate rays (dashed line in Fig. 4). Als, 
7 E. wl 
let the unit vectors corresponding to T;, and T» be represented by T , and T , respectively, then from the vector 
polygon in Fig 4. we can write the vector equation 
P=b-sT+s,T, 7) 
We employ the least squares optimization technique to determine values for s,, s» which minimize the length of vecto 
p. For this we formulate a vector space functional involving P whose stationary values yield the required values fors, Th 
and s». Using the inner product vector functional we have : pr 
mi 
o(P.P) 
e ed 
Os, 
o(PP or 
en, 
os, 
substituting for p from (7) provides 
Q(b — sTi s,T2).(b — sTi s,T2) zt an 
Os, = ve 
O(b—-sT1+s,T2)(b—s, T1 + s, T2) <0 P 
os, 
Employing vector differential operators (Olaleye 1992), we obtain the normal equations from (8) as 
  
664 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.